nLab quasi-finite CW-complex

Definition

A countable CW-complex $K$ is quasi-finite if for any finite subcomplex $M\subset K$, there is (possibly larger) finite subcomplex $e(M)\subset K$, such that for every separable metric space $X$ satisfying

• ($K$ is an absolute extensor of $X$:) for every closed subspace $A\subset X$ and a function $f:A\to K$ there is an extension $\tilde{f}:X\to K$ (i.e. $\tilde{f}=f\circ i$, where $i:A\hookrightarrow X$ is the closed embedding)

one has a similar property

• for every closed subspace $A\subset X$ and a function $f:A\to M$ there is an extension $g:X\to e(M)$ (i.e. $g\circ i = f$).

There is a characterization: a coutable CW-complex $K$ is quasi-finite iff for all separable metric spaces $X$, if $K$ is an absolute extensor of $X$ implies then it is an absolute extensor of its Stone-Čech compactification $\beta(X)$ as well.

In fact, the original definition asks that one has a function $e:M\to e(M)$ (the same under the axiom of choice).

References

• A.Karasev, On two problems in extension theory, arXiv:math.GT/0312269

• M.Cencelj, J.Dydak, J.Smrekar, A.Vavpetic, Ž.Virk, Algebraic properties of quasi-finite complexes, Fund. Math. 197 (2007), 67-80 math/0509582